Fisher Information for Inverse Problems and Trace Class Operators

This paper provides a mathematical framework for Fisher information analysis for inverse problems based on Gaussian noise on infinite-dimensional Hilbert space. The covariance operator for the Gaussian noise is assumed to be trace class, and the Jacobian of the forward operator Hilbert-Schmidt. We show that the appropriate space for defining the Fisher information is given by the Cameron-Martin space. This is mainly because the range space of the covariance operator always is strictly smaller than the Hilbert space. For the Fisher information to be well-defined, it is furthermore required that the range space of the Jacobian is contained in the Cameron-Martin space. In order for this condition to hold and for the Fisher information to be trace class, a sufficient condition is formulated based on the singular values of the Jacobian as well as of the eigenvalues of the covariance operator, together with some regularity assumptions regarding their relative rate of convergence. An explicit example is given regarding an electromagnetic inverse source problem with "external" spherically isotropic noise, as well as "internal" additive uncorrelated noise.Comment: Submitted to Journal of Mathematical Physic

Shape optimization for 3D electrical impedance tomography

In the present paper we consider the identification of an obstacle or void of different conductivity included in a three-dimensional domain by measurements of voltage and currents at the boundary. We reformulate the given identification problem as a shape optimization problem. Since the Hessian is compact at the given hole we apply a regularized Newton scheme as developed by the authors (WIAS-Preprint No. 943). All information of the state equation required for the optimization algorithm can be derived by boundary integral equations which we solve numerically by a fast wavelet Galerkin scheme. Numerical results confirm that the proposed regularized Newton scheme yields a powerful algorithm to solve the considered class of problems

Monotonicity in inverse scattering for Maxwell’s equations

We consider the inverse scattering problem to recover the support of penetrable scattering objects in three-dimensional free space from far field observations of scattered time-harmonic electromagnetic waves. The observed far field data are described by far field operators that map superpositions of plane wave incident fields to the far field patterns of the corresponding scattered waves. We discuss monotonicity relations for the eigenvalues of linear combinations of these operators with suitable probing operators. These monotonicity relations yield criteria and algorithms for reconstructing the support of scattering objects from the corresponding far field operators. To establish these results we combine the monotonicity relations with certain localized vector wave functions that have arbitrarily large energy in some prescribed region while at the same time having arbitrarily small energy on some other prescribed region. Throughout we suppose that the relative magnetic permeability of the scattering objects is one, while their real-valued relative electric permittivity maybe inhomogeneous and the permittivity contrast may even change sign. Numerical examples illustrate our theoretical findings

B-spline based sharp feature preserving shape reconstruction approach for electrical impedance tomography

This paper presents a B-spline based shape reconstruction approach for electrical impedance tomography (EIT). In the proposed approach, the conductivity distribution to be reconstructed is assumed to be piecewise constant. The geometry of the inclusions is parameterized using B-spline curves, and the EIT forward solver is modified as a set of control points representing the inclusions’ boundary to the data on the domain boundary. The low order representation decreases the computational demand and reduces the ill-posedness of the EIT reconstruction problem. The performance of the proposed B-spline based approach is tested with simulations which demonstrate the most popular biomedical application of EIT: lung imaging. The approach is experimentally validated using water tank data. In addition, robustness studies of the proposed approach considering varying initial guesses, inaccurately known contact impedances, differing numbers of control points, and degree of B-spline are performed. The simulation and experimental results show that the B-spline based approach offers improvements in image quality in comparison to the traditional Fourier series based reconstruction approach, as measured by quantitative metrics such as relative size coverage ratio and relative contrast. Inasmuch, the proposed approach is demonstrated to offer clear improvement in the ability to preserve the sharp properties of the inclusions to be imaged

3D Reconstruction for Partial Data Electrical Impedance Tomography Using a Sparsity Prior

In electrical impedance tomography the electrical conductivity inside a physical body is computed from electro-static boundary measurements. The focus of this paper is to extend recent result for the 2D problem to 3D. Prior information about the sparsity and spatial distribution of the conductivity is used to improve reconstructions for the partial data problem with Cauchy data measured only on a subset of the boundary. A sparsity prior is enforced using the $\ell_1$ norm in the penalty term of a Tikhonov functional, and spatial prior information is incorporated by applying a spatially distributed regularization parameter. The optimization problem is solved numerically using a generalized conditional gradient method with soft thresholding. Numerical examples show the effectiveness of the suggested method even for the partial data problem with measurements affected by noise.Comment: 10 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1405.655

DICOM for EIT

With EIT starting to be used in routine clinical practice [1], it important that the clinically relevant information is portable between hospital data management systems. DICOM formats are widely used clinically and cover many imaging modalities, though not specifically EIT. We describe how existing DICOM specifications, can be repurposed as an interim solution, and basis from which a consensus EIT DICOM ‘Supplement’ (an extension to the standard) can be writte